Find The Value Of $c$ That Completes The Square For The Expression $p^2 + 34p + C$.

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Introduction to Completing the Square

Completing the square is a powerful algebraic technique used to rewrite quadratic expressions in a specific form. This method involves manipulating the expression to create a perfect square trinomial, which can be factored into the square of a binomial. In this article, we will focus on finding the value of c that completes the square for the expression p^2 + 34p + c.

Understanding the Concept of Completing the Square

To complete the square, we need to create a perfect square trinomial from the given quadratic expression. A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. The general form of a perfect square trinomial is (a + b)^2 = a^2 + 2ab + b^2.

The Formula for Completing the Square

The formula for completing the square involves taking half of the coefficient of the linear term, squaring it, and adding it to both sides of the equation. In this case, the coefficient of the linear term is 34. Half of 34 is 17, and 17^2 is 289.

Applying the Formula to the Given Expression

To complete the square for the expression p^2 + 34p + c, we need to add 289 to both sides of the equation. This will create a perfect square trinomial on the left-hand side of the equation.

p^2 + 34p + c = (p + 17)^2

Expanding the Perfect Square Trinomial

To expand the perfect square trinomial, we need to square the binomial (p + 17). This will give us p^2 + 34p + 289.

(p + 17)^2 = p^2 + 34p + 289

Equating the Two Expressions

Now that we have expanded the perfect square trinomial, we can equate it to the original expression.

p^2 + 34p + c = p^2 + 34p + 289

Solving for c

To solve for c, we need to isolate c on one side of the equation. We can do this by subtracting p^2 + 34p from both sides of the equation.

c = 289

Conclusion

In this article, we have found the value of c that completes the square for the expression p^2 + 34p + c. The value of c is 289. This value allows us to rewrite the expression as a perfect square trinomial, which can be factored into the square of a binomial.

Examples of Completing the Square

Completing the square is a powerful technique that can be used to solve quadratic equations and rewrite quadratic expressions in a specific form. Here are a few examples of completing the square:

Example 1

Find the value of c that completes the square for the expression x^2 + 12x + c.

To complete the square, we need to add 36 to both sides of the equation.

x^2 + 12x + c = (x + 6)^2

Expanding the perfect square trinomial, we get:

x^2 + 12x + c = x^2 + 12x + 36

Solving for c, we get:

c = 36

Example 2

Find the value of c that completes the square for the expression y^2 + 20y + c.

To complete the square, we need to add 100 to both sides of the equation.

y^2 + 20y + c = (y + 10)^2

Expanding the perfect square trinomial, we get:

y^2 + 20y + c = y^2 + 20y + 100

Solving for c, we get:

c = 100

Applications of Completing the Square

Completing the square has many applications in mathematics and other fields. Here are a few examples:

Quadratic Equations

Completing the square can be used to solve quadratic equations. By rewriting the quadratic expression as a perfect square trinomial, we can factor it into the square of a binomial and solve for the variable.

Conic Sections

Completing the square can be used to rewrite conic sections in a specific form. This can be useful for graphing and analyzing conic sections.

Statistics

Completing the square can be used to find the minimum or maximum value of a quadratic function. This can be useful in statistics and data analysis.

Conclusion

In this article, we have found the value of c that completes the square for the expression p^2 + 34p + c. We have also discussed the concept of completing the square, the formula for completing the square, and the applications of completing the square. Completing the square is a powerful technique that can be used to rewrite quadratic expressions in a specific form and solve quadratic equations.

Introduction

Completing the square is a powerful algebraic technique used to rewrite quadratic expressions in a specific form. In our previous article, we discussed the concept of completing the square, the formula for completing the square, and the applications of completing the square. In this article, we will answer some frequently asked questions about completing the square.

Q&A

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I complete the square for a quadratic expression?

A: To complete the square for a quadratic expression, you need to follow these steps:

  1. Take half of the coefficient of the linear term and square it.
  2. Add the result to both sides of the equation.
  3. Simplify the expression to create a perfect square trinomial.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

a^2 + 2ab + b^2 = (a + b)^2

Q: How do I apply the formula to a quadratic expression?

A: To apply the formula to a quadratic expression, you need to follow these steps:

  1. Identify the coefficient of the linear term.
  2. Take half of the coefficient and square it.
  3. Add the result to both sides of the equation.
  4. Simplify the expression to create a perfect square trinomial.

Q: What are the applications of completing the square?

A: Completing the square has many applications in mathematics and other fields, including:

  • Solving quadratic equations
  • Graphing conic sections
  • Finding the minimum or maximum value of a quadratic function
  • Data analysis and statistics

Q: Can I use completing the square to solve quadratic equations?

A: Yes, completing the square can be used to solve quadratic equations. By rewriting the quadratic expression as a perfect square trinomial, you can factor it into the square of a binomial and solve for the variable.

Q: How do I use completing the square to solve quadratic equations?

A: To use completing the square to solve quadratic equations, you need to follow these steps:

  1. Rewrite the quadratic expression as a perfect square trinomial.
  2. Factor the perfect square trinomial into the square of a binomial.
  3. Solve for the variable.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

  • Not taking half of the coefficient of the linear term
  • Not squaring the result
  • Not adding the result to both sides of the equation
  • Not simplifying the expression to create a perfect square trinomial

Q: Can I use completing the square to rewrite conic sections?

A: Yes, completing the square can be used to rewrite conic sections. By rewriting the conic section as a perfect square trinomial, you can analyze and graph it more easily.

Q: How do I use completing the square to rewrite conic sections?

A: To use completing the square to rewrite conic sections, you need to follow these steps:

  1. Rewrite the conic section as a quadratic expression.
  2. Complete the square to create a perfect square trinomial.
  3. Rewrite the conic section in the form of a perfect square trinomial.

Conclusion

In this article, we have answered some frequently asked questions about completing the square. Completing the square is a powerful technique used to rewrite quadratic expressions in a specific form and solve quadratic equations. By understanding the concept of completing the square and applying the formula correctly, you can use completing the square to solve a wide range of problems in mathematics and other fields.

Additional Resources

If you want to learn more about completing the square, here are some additional resources:

  • Khan Academy: Completing the Square
  • Mathway: Completing the Square
  • Wolfram Alpha: Completing the Square

Practice Problems

To practice completing the square, try solving the following problems:

  • Complete the square for the expression x^2 + 12x + c.
  • Complete the square for the expression y^2 + 20y + c.
  • Use completing the square to solve the quadratic equation x^2 + 6x + 8 = 0.
  • Use completing the square to rewrite the conic section x^2 + 4y^2 + 2x + 8y = 0.

I hope this helps! Let me know if you have any questions or need further clarification.